Computational assessment of Sn activities and integral excess free energy change for mixing in the Sn-Au-Cu ternary liquid alloys using the molecular interaction volume model

The activities of Sn in the liquid solder ternary alloy Sn-Au-Cu at 900 K have been computed using the molecular interaction volume model (MIVM). The calculated values have been compared with the experimental data for three cross-sections, i.e., for three different ratios of aurum to copper (XAu:XCu) = 3:1, 1:1, and 1:3. In addition, the excess Gibbs free energy of mixing, ΔGEx, for these ternary liquid alloys has been determined using the same model parameters to assess their validity. The resulting values have then been compared with the corresponding experimental data found in the literature. The agreement between the theoretical and experimental results has been found to be satisfactory.


Introduction
The investigation of the thermodynamic characteristics of different alloys can be accomplished through experimental measurements.However, conducting these experiments often requires significant time and financial resources.Consequently, obtaining all the necessary thermodynamic data through experiments presents challenges.Thus, exploring theoretical methods to acquire this data would be more favorable [1].Traditionally, lead-based systems have been extensively used for soldering purposes due to their costeffectiveness.Nonetheless, the use of lead in soldering poses environmental and health risks as it is a toxic metal [2][3][4][5].As a result, global regulations have banned the use of lead in soldering.It's worth noting that replacing lead in soldering technology is a more cost-effective approach than attempting to manage electronic garbage.Consequently, the development of lead-free solders has become a critical focus in the electronics industry [3].
For the new lead-free solder alloys to be viable, they must possess similar characteristics like low melting temperatures, wetting characteristics, rust resistance, resistance to fatigue, being non-poisonous, etc.Hence, researchers have made various efforts to find suitable replacements for lead in solder alloys [3].
Tin (Sn) is a gentle, white-silver metallic element known for its excellent corrosion resistance due to its limited tendency to oxidize.Tin is generally considered non-toxic to most organisms.Its relatively low melting point makes it a preferred material for manufacturing solder [6].Gold (Au) possesses a yellow color, is easily malleable, and exhibits ductility.Gold alloys have very high tensile and shear strength, high thermal conductivity, and thermal fatigue.It flows well in a molten state.The main properties of copper (Cu) are that it is malleable, ductile, and corrosion-resistant.Compared to other materials, copper is a superior conductor, less costly, and less magnetic.Au-Cu-Sn alloys possess unique properties that make them suitable for specific applications.These alloys can be designed or modified to have specific properties such as improved electrical conductivity, corrosion resistance, or mechanical strength that are not readily available in existing materials.These alloys may improve solder joint reliability and performance, and they may also be suitable for microelectronic packaging and interconnects.As they are novel, more environmentally friendly, or easier to recycle, we are motivated to investigate Au-Cu-Sn alloys.
The main goal in alloy thermodynamics is to determine the thermodynamic activities of components within alloys.This is crucial for comprehending metallurgical processes and observing how chemical interactions occur between alloy systems and standard base materials [7][8][9][10][11][12].Au-Sn, Sn-Ag, Sn-Cu, Al-Sn-Zn, Sn-Ag-Cu, Sn-Sb-Bi, Zn-In-Sn, Sn-Ag-Cu, In-Bi-Sn, Sn-Ag-Cu-Zn, Sn-Ag-Cu-Sb, and other combinations have all proven to be effective alternatives to traditional Sn-Pb solder alloys [2,6,[13][14][15][16].In the prediction of thermodynamic properties within ternary and more complex systems, a wide range of geometric models, including but not limited to Kohler, Muggianu, Collinet, Chou, the Kaptay model, R-K equations, and others [17,18], find wide application.The tin activity in the liquid Au-Cu-Sn alloys was measured using solid electrolyte galvanic cells and an emf technique by Wierzbicka-Miernik et al. [19].The Au-Cu-Sn ternary system was thermodynamically reassessed on the basis of the experimental results from the work of Dong et al., as well as the data available in the literature and the thermodynamic descriptions of the binary systems in 2014 [20].The thermodynamic properties of the Au-Cu-Sn ternary system were assessed using the electromotive force (EMF) method with a liquid electrolyte at three distinct cross-sections.These cross-sections corresponded to constant Au:Cu ratios of 3:1, 1:1, and 1:3, and the measurements were conducted within the temperature range spanning from the liquidus temperature of the alloys up to 1023 K [21].A geometrical model has been widely used in the property prediction of melts on binary systems, and then the optimized geometrical model (OGM) is verified in predicting the physicochemical properties of Sn-based ternary alloys Sn-Au-Cu [22].However, their excess free energy of mixing, ΔG Ex , and the components' activity are not present in the literature using the molecular interaction volume model (MIVM).Hence, the MIVM has been used in this study to calculate Sn's activity in liquid ternary alloys (Sn-Au-Cu) at 900 K.We have concentrated on three cross-sections with various X Au /X Cu ratios, namely 3:1, 1:1, and 1:3.Furthermore, the theoretical model has been utilized to calculate ΔG Ex of Sn-Au-Cu at 900 K for the variable concentration of Sn (X Sn ) at the three cross-sections of X Au :X Cu = 3:1, 1:1, and 1:3.

Theory
The MIVM (Molecular Interaction Volume Model) investigates the properties of liquid molecule migration from a physical perspective.The model suggests that liquid molecules differ from gas molecules, which constantly move in an irregular manner, as well as from solid molecules, which oscillate continuously in one place but shift in a non-random manner between molecular cells.In this model, the central molecules and their closest counterparts can be exchanged, and the cells they occupy are both movable and indistinguishable [23].The thermodynamic properties of the components of a multicomponent mixture can be predicted using a twoparameter model that only requires the regular physical quantities of pure liquid metals and the associated binary infinite dilute activity coefficient [24].
The integral excess free energy, ΔG Ex , of a binary liquid alloy is expressed as [25].
In this context, V and V mi mj represent the molecular volumes, while x i and x j denote the concentrations of constituents i and j in a binary liquid alloy.The interaction parameters that describe the pair's potential energy between the constituents have been denoted by A ji and A ij .Z i and Z j represent the initial coordination numbers of the constituent metals i and j of the binary liquid alloy.T and R stand for the absolute temperature and the gas constant, respectively.
The ith and jth components' activity coefficients in a liquid binary system, denoted as g i and g , j are expressed as follows, according to Tao [25]: When considering a multicomponent mixture, which involves systems of higher orders, equation (1) can be transformed into the form given by [26].
This is referring to the variable 'N', which represents the total number of components present in the mixture.
The formula for calculating the ith component's activity coefficient (g i ) is provided by Tao [26].
The expression for the first coordination number Z i is [23]: Here, in the formula for molecular number density, r = N V , i i mi N i represents the molecular number whose value is taken as 0.66022.∆H mi represents the enthalpy of melting, and T mi represents the melting point of the metal.Z c represents the close-packed coordination for liquid metals, with a value of 12 in the present study.Meanwhile, r oi corresponds to the first peak value of the radial distribution function, and r mi stands for the initial value of the radial distribution function.The pair's potential energy's interaction parameters (A ji and A ij ) are given by [15].
In the given expression, 'e ji ' refers to the potential energy between a central i atom and its first nearby j atom.Additionally, K stands for the Boltzmann constant.The coefficients of activity (g ¥ i ) and (g ¥ j ) for the ith and jth components of the liquid binary system correspond to the infinite dilute range and are expressed as follows in Tao's work [25]: The Newton-Raphson method is used to solve equations ( 8) and (9) to determine the values for A ji and A ij .The results of this computation serve as starting points for the computation of component activity in binary liquid alloys using equations (2) and (3).If the suitable A ji and A ij values are once determined at a certain temperature, their equivalent values at subsequent temperatures can likewise be calculated, as described in Tao's work [25].
Now, the activity coefficient of the first constituent metal in the ternary liquid alloy Sn-Au-Cu (denoted as g 1 ) can be calculated using equation (5) [25].Likewise, the integral excess free energy of mixing (ΔG Ex ) for Sn-Au-Cu ternary liquid alloys can be determined using equation (4) [25]. )

Results and Discussion
Table 1 contains certain input variables, while Table 2 provides the values of some estimated parameters of the binary systems.By substituting the values of V m1 , V m2 , V m3 , Z 1 , Z 2 , Z 3 , A 12 , A 21 , A 13 , A 31 , A 23 , and A 32 into equations (10) and (11), the component Sn activities in the Sn-Au-Cu system at 900 K and the integral excess free energy (ΔG Ex ) of mixing the ternary liquid alloys at 900 K have been calculated.These computed values are then compared with the corresponding experimental values [21], presented in Tables 3 and 4. Additionally, the results are visually illustrated in figures 1, 2, and 3, which represent the three cross-sections of X Au :X Cu = 3:1, 1:1, and 3:1, respectively.The average standard deviation ( * S i ) and average relative error (S i ) have been derived from Table 3 in order to precisely measure the extent of the difference between experimental values and estimated data.Equation (12) can be used to get the average standard deviation.
Equation (13) can be used to get the average relative error.

100
, Th. , Exp. , Exp. 13 The current study has taken 31 experimental data points (n) into account.It is found from Table 3 that the values of the standard deviation and average relative error as determined by MIVM are 0.0276% and 12.95%, The agreement between the theoretical values of Sn's activities in Sn-Au-Cu liquid alloys at 900 K and the experimental data, as observed in the study conducted by Hindler et al. [21], is noticeable in Table 3.However, some discrepancies exist, with maximum errors of 57.51% at X Sn = 0.2021 for the cross-section X Au :X Cu = 3:1, 48.51% at X Sn = 0.2014 for the cross-section X Au :X Cu = 1:1, and 41.36% at X Sn = 0.2004 for the cross-section X Au :X Cu = 1:3, respectively.Here, too, it is clear that there is a higher percentage deviation of the theoretical activity of Sn than that of the experimental activity of Sn at lower concentrations of Sn, i.e., X Sn , in the Sn-Au-Cu liquid system for all three cross-sections, i.e., X Au :X Cu = 3:1, 1:1, and 1:3.
In addition, Table 4 illustrates that the nature of theoretical variations of ΔG Ex of the Sn-Au-Cu liquid system at 900 K, as observed in the experimental study conducted by Hindler et al. [21], is almost similar for all three different cross-sections, i.e., X Au :X Cu = 3:1, 1:1, and 1:3.
Figure 1(a) shows that the predicted Sn's activity in Sn-Au-Cu alloys at 900 K follows the same pattern as the experimental activity [21] for the cross-section X Au :X Cu = 3:1, except in the concentration range 0.7 x Sn 0.9.The Sn activity deviation from the ideal Raoult's line for this cross-section is negative.A departure from Rauolt's law in a negative direction signifies that the observed vapor pressure is lower than anticipated.This phenomenon * S i = 0.0276% , S i = 12.95%.occurs when the intermolecular forces between Sn and other components, such as Au and Cu, are stronger than the average intermolecular forces found in the pure components of the mixture.The plot of ΔG Ex versus X Sn for Sn-Au-Cu liquid alloys at 900 K for the cross-section X Au :X Cu = 3:1, as shown in figure 1(b), also exhibits a similar tendency between the theory and experiment [21].ΔG Ex consistently exhibits negative values across the full range of concentrations.The most pronounced negative values are −16220.1 J mol −1 (Th.) at X Sn = 0.3 and −9062.0J mol −1 (Exp.) at X Sn = 0.4.Figure 2(a) also shows that the predicted Sn's activity in Sn-Au-Cu alloys at 900 K follows the same pattern as the experimental activity [21] for the cross-section X Au :X Cu =1:1, except in the concentration range 0.7 x Sn 0.9.The Sn activity deviation from the ideal Raoult's line is large negative in the lower concentrations of Sn and slight positive at the higher concentrations of Sn for this cross-section.A negative deviation from Raoult's law indicates that the observed vapour pressure of Sn in the mixture is lower than the expected value, while a positive deviation indicates that the observed vapour pressure is greater than the expected value in the case of an ideal mixture.The plot of ΔG Ex versus X Sn for Sn-Au-Cu liquid alloys at 900 K for the cross-section X Au :X Cu =1:1, as shown in figure 2(b), also exhibits a similar tendency between theory and experiment [21].ΔG Ex consistently exhibits negative values across the full range of concentrations.The most pronounced negative values are −19365.5J mol −1 (Th.) at X Sn = 0.2 and −7841.0J mol −1 (Exp.) at X Sn = 0.3.
From figure 3(a), it is evident that the predicted Sn's activity in Sn-Au-Cu alloys at 900 K for the cross-section X Au :X Cu =1:3 also follows the same pattern as the experimental activity [21], except in the concentration range 0.6 x Sn 0.8 and also at X Sn = 0.2.The deviation of Sn activity from the ideal Raoult's line is negative at the lower concentrations of Sn and slightly positive at the higher concentrations of Sn for this cross-section.Here, a negative deviation from Raoult's line indicates that the observed vapour pressure of Sn in the combination is  lower than the expected value, while a positive deviation indicates that the observed vapour pressure is greater than it would be in an ideal system.The graphical plot of ΔG Ex versus X Sn for Sn-Au-Cu liquid alloys at 900 K for the cross-section X Au :X Cu =1:3, as shown in figure 3(b), also exhibits a similar tendency between the theory and experiment [21].There are negative values of ΔG Ex , with maximum negative values of −17719.6J mol −1 (theoretical) and −5251.0J mol −1 (experimental), both at X Sn = 0.2.A negative value of ΔG Ex of the liquid ternary alloys at all three cross-sections of Sn-Au-Cu at 900 K indicates that the actual free energy of the mixture is lower than what would be expected if the components mixed perfectly according to ideal behavior.This suggests that there are attractive interactions between the different components of the alloy that are stronger than the average interactions in the pure components.Furthermore, the negative excess free energy implies that the components of the ternary alloy tend to mix more favorably and are somewhat attracted to each other, leading to a lower overall free energy of the mixture compared to a completely ideal mixture.This can be a result of various factors, such as chemical bonding, intermolecular forces, or specific interactions between the components in the alloy.
As can be observed in figures 1(b), 2(b), and 3(b), there is a more apparent divergence between experimental results and forecasts, despite the similarity of the curves' characteristics.The determined values of the model parameters A ij and A ji , which are the best fitting values, may be the cause of the observed inconsistencies [6].It is noteworthy to note that, due to their dependency on composition and temperature, all thermodynamic features cannot be adequately explained by a theoretical model simultaneously.As an illustration, consider the formula: ΔG = ΔH -T S = f (x, T), where ΔH, ΔG, and ΔS stand for, respectively, the enthalpy change, change in Gibbs free energy, and entropy change of the liquid system [28].
Theoretical models are often built upon a set of assumptions and simplifications that may not accurately reflect the real-world scenario.In contrast, experimental results are obtained through real-world observations that take into account all the complexities of the system under investigation.Therefore, it is possible for theoretical results to diverge from experimental results because of the oversimplification of the system in the theoretical model.Molecular interaction models typically assume that the system is in a particular phase (e.g., gas, liquid, or solid) under specified conditions.Deviations can occur if the actual system is in a different phase or exhibits phase transitions that are not accounted for in the model.Here, it should be noted that the liquidus temperature of Sn-Au-Cu alloys is about 900 K; however, the melting points of each component are different.Sometimes, there may be some unknown external factors that influence the experiment that are not considered in the theoretical model.These are the reasons that could account for the significant disparities observed between the theoretical and experimental data concerning the Sn activity in Sn-Au-Cu liquid alloys at 900 K, as well as their excess Gibbs free energy of mixing.However, the model has the benefit of being able to estimate the thermodynamic characteristics of binary, ternary, and quaternary liquid alloys using only binary infinite dilute activity coefficients.The current model offers at least the qualitative nature of ΔG Ex for the Sn-Au-Cu system at 900 K with respect to X Sn .For the liquid Sn-Au-Cu system at 900 K, there are no more studies using the MIVM on ΔG Ex .The projected ΔG Ex values for ternary liquid alloys in this study will therefore be useful for a more detailed explanation of the thermodynamic characteristics of Sn-Au-Cu liquid alloys.

Conclusions
Theoretical examination indicates that the molecular interaction volume model (MIVM) is a suitable approach for describing the thermodynamic activities of the constituents in ternary liquid alloys and their excess free energy of mixing (ΔG XS ).The study successfully explains the negative deviation of Sn activity from ideal behavior in the entire concentration range of Sn for the cross-section X Au :X Cu = 3:1 in Sn-Au-Cu liquid alloys at 900 K. Additionally, the study accounts for the negative deviation of Sn activity from ideal behavior at lower Sn concentrations and a slightly positive deviation at higher Sn concentrations for both cross-sections of X Au :X Cu = 1:1 and 1:3 in Sn-Au-Cu liquid alloys at 900 K.
To assess the accuracy of the model, the study has compared the integral excess free energy of the Sn-Au-Cu system, determined by Duhem integration, with the calculations made using the MIVM model.It's worth noting that there are no other studies available on the ΔG XS of Sn-Au-Cu liquid alloys using the MIVM model.Consequently, the theoretical data for ΔG XS provided in this work will be valuable for more precisely explaining the thermodynamic properties of Sn-Au-Cu liquid alloys.It is possible that research into the thermodynamic characteristics of liquid Sn-Au-Cu alloys will result in the development of novel materials, improved manufacturing processes, increased environmental responsibility, improved performance in electronic devices, and advancements in other crucial technological fields.

Figure 1 .
Figure 1.Theoretical and experimental [21] variation of (a) activity of Sn versus X Sn and (b) ΔG Ex versus X Sn for the Sn-Au-Cu liquid system at 900 K at cross-section X Au :X Cu = 3:1.

Figure 2 .
Figure 2. Theoretical and experimental [21] variation of (a) activity of Sn versus X Sn and (b) ΔG Ex versus X Sn for the Sn-Au-Cu liquid system at 900 K at cross-section X Au :X Cu = 1:1.

Figure 3 .
Figure 3. Theoretical and experimental [21] variation of (a) activity of Sn versus X Sn and (b) ΔG Ex versus X Sn for the Sn-Au-Cu liquid system at 900 K at cross-section X Au :X Cu = 1:3.

Table 2 .
Evaluated Z i , Z j , A ij , and A ji for various binary systems at various temperatures.
respectively, which are almost lesser values.This supports the model's accuracy in estimating the activity of the Sn component in the Sn-Au-Cu ternary liquid alloy at 900 K.